Theorem stge0

Description: The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	stge0	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → 0 ≤ ( 𝑆 ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sticl	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) )
2		elicc01	⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝑆 ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( 𝑆 ‘ 𝐴 ) ∧ ( 𝑆 ‘ 𝐴 ) ≤ 1 ) )
3	2	simp2bi	⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) → 0 ≤ ( 𝑆 ‘ 𝐴 ) )
4	1 3	syl6	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → 0 ≤ ( 𝑆 ‘ 𝐴 ) ) )